Source Term Parameterization for PCA Combustion Modeling
|
|
- Asher Craig
- 5 years ago
- Views:
Transcription
1 Paper # 7RK-252 Topic: Reaction Kinetics 8 th US ational Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 9-22, 23. Source Term Parameterization for PCA Combustion Modeling Isaac, B.,2 Parente, A. 2 Sutherland, J. Smith, P. Fru, G. 3 Thevenin D. 3 Chemical Engineering Department, University of Utah, Salt Lake City, Utah 2 Aero-Thermo-Mechanique, Universite Libre de Bruxelles, Brussels, Belgium 3 Institute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke University Magdeburg, Magdeburg, Germany Modeling the physics of turbulent combustion systems remains a challenge due to large range of scales, which are important in these systems. Often, detailed chemical kinetic mechanisms are used to fully describe the chemistry involved in the combustion process, yielding highly coupled partial differential equations for each of the chemical species used in the mechanism. Recently, Principal Components Analysis (PCA) has shown promise in its ability to identify a low dimensional manifold describing the reacting system []. Sutherland and Parente demonstrated the formulation of a PCA model [2] where the Principal Components (PCs) of the system are transported. Evaluation of the PCs source-term is expensive (as all chemical species source-terms must be evaluated) and inherits error through the PC approximation. Parameterization methods can be employed to quickly and accurately produce sourceterm values, allowing one to avoid the expensive calculation of the source-terms and avoid the additional error from the state space approximation. The present work demonstrates the ability to parameterize the source-term space of a 2D Direct umerical Simulation of a spherical premixed syn-gas (CO/H 2 ) air flame using non-linear regression, comparing several non-linear regression methods. In addition the ability to parameterize the source-terms while altering the scaling parameters used in PCA is interesting as the scaling greatly effects the behavior and shape of the low-dimensional space being modeled. Introduction The ability to accurately model a turbulent combustion system remains challenging due to the complex nature of combustion systems. A simple fuel such as CH 4 has been accurately described using 53 species and 325 chemical reactions [3]. More complex fuels require increasingly complex chemical mechanisms. Each resolved chemical species requires a conservation equation which is a coupled, highly non-linear partial differential equation. Such systems are only possible to solve under very limited situations at this time due to computational costs. This issue leads to the need of a reduced model, which can adequately describe the chemical reactions. Many methods such as computational singular perturbation (CSP) [4], and Rate Controlled Constrained Equilibrium (RCCE) [5] attempt to reduce the complexity of the mechanism by using equilibrium assumptions for fast chemical processes, and using the computational resources on the more pertinent evolution
2 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics in the reaction process. Indeed in these complex combustion reaction mechanisms many of the species evolve at time scales much smaller than the time scales of interest, allowing for decoupling of fast and slow processes while maintaining accuracy. Low dimensional manifolds exists in these systems which describe well the governing characteristics of the flames. For example, the steady laminar flamelet model [6] uses the mixture fraction and mixture fraction variance to describe the flame as an ensemble of steady laminar diffusion flames undergoing various strain rates, providing a very good representation of the entire system with low number of variables. Principal Components Analysis (PCA) has been shown to identify the low dimensional manifolds [7] for turbulent combustion systems. PCA uses the eigenvalue decomposition of the covariance matrix of the thermodynamic state space to identify the manifold. In previous work by Sutherland [7], a modeling approach was presented which uses conservation equations for q PCs (η) which are calculated from q independent linear combinations of the state space variables. The selection of q depends on the complexity of the system of interest, as well as the desired accuracy of the representation of the system. The transport equations for η are of the following form: t (ρη)+ (ρu i η)= j η + s η () x i x i where j η is the diffusive flux of η and s η is the source-term for η. A major challenge of this modeling strategy is in the evaluation of s η. Introduction of a small amount of error to the state space variables by using a PCA representation, can dramatically effect the chemical species source-term (ω k ) calculation. The error in many cases is exponentially propagated due to the characteristics of the reaction rate equations. The present work investigates the ability to accurately model s η using a high-fidelity data set containing exact or similar physics to the system of interest. on-linear regression is used to create a model for s η as a function of η, where the training values for s η are calculated from a training set of X which contain no error from PCA approximation. The benefit in this approach is that approximation error due to PCA on the system is not propagated into the model for s η. Several well known non-linear regression techniques are investigated for estimation of s η, as well as a novel regression method based on Gaussian kernels filters. In addition the effects of the scaling used in PCA is assessed in terms of the ability to create regressions models for s η. 2 Approach 2. Principal Component Analysis The PCs (η) are calculated from the PCA analysis. PCA is performed on a data set consisting of n observations with k variables organized as an n k matrix (X). The data X is centered to zero by its corresponding means X, and scaled by the diagonal matrix γ containing the scaling value for each of the k variables: X =(X X)γ (2) PCA identifies a basis matrix A which when multiplied with X creates an approximation to η. The accuracy of the approximation of η is dependent on the number of retained columns of A. A is obtained by the eigenvalue decomposition of the covariance matrix of X: 2
3 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics k X X = A ΛA (3) The PCs are then defined by the projection of the basis matrix onto the scaled and centered data η = XA (4) The amount of variance represented by basis matrix columns are ordered from highest to lowest. Accordingly a subset of q columns of the original k columns in A, where q k may yield a good approximation of X. Accordingly a subset of η may be used to approximate the entire state space with minimal error based on the number of retained eigenvectors from A. X ηa T kq (5) The present application of PCA to the turbulent combustion system uses PCA to approximate the k chemical species. 2.2 PCA Scaling Scaling plays a key role in PCA as well as the non-linear regression that follows. After centering X the data needs to be scaled so that the PCA will give equal weights to the independent variables (γ from Equation 2). The following scaling methods where adopted for this study [8]: - auto scaling (), uses the standard deviation s k. Auto scaling leaves all columns of X with a standard deviation of one, and now the data is analyzed on the basis of correlations instead of covariances, γ k = s k. - range scaling (RA), uses the difference between the minimum and the maximum variable value, γ k =max X k X k min Xk X k. - pareto scaling (PAR), adopts the square root of the standard deviation as scaling factor, γ k = s k. - variable stability scaling (), gives an emphasis to variables which do not show strong variation, by using the product between the standard deviation and the coefficient of variation, γ k = s k k X k s. - level scaling (LEV), uses the mean value of the variables γ k = X k. - max scaling (), uses the maximum variable value as the scaling factor, here γ k = max X k X k. The scaling of the data set effects the shape of the low dimensional manifold calculated from PCA, which yields a significant impact on the ability of the non-linear regression (see Section 3). 3
4 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 2.3 Principal Components Conservation Equation As is discussed in the work by Sutherland [7], the conservation equations for the PCs are derived from the general species transport equation [9]: t (ρy k)+ (ρu i Y k )= Y k ρd k + ω k (6) x i x i x i PCA provides a kq, the scaling vector γ k, and the centering vector Ȳk. One derives the transport equations for η by first centering and scaling the species mass fractions: ρ Y k Ȳk + Y k ρu Ȳk i = t γ k x i γ k x i multiplying by a kq and substituting Equation 4 leaves: t (ρη)+ (ρu i η)= x i s η = x i Substituting the diffusion term for Ficks law yields Equation. D η D k x i Yk Ȳk γ k + ω k ργ k (7) (η) + s η (8) x i ω k ργ k a kq (9) The source-term for this Equation (s η ) is a highly non-linear function of all of the state variables. Although the resolution of this source-term should be straight forward as it is a simply a function of the species mass fractions and the temperature, issues arise due to the approximated state space. The error in the approximation of the state space propagates into s η. A non-linear regression model can be used to model this source-term as a function of η. By training the non-linear function on values of s η that are free from the PCA approximation errors, the regression will provide accurate values for s η even though the state space is approximated. 2.4 Regression Models In this study on-linear Regression models are used to develop a function, f, which estimates the source-terms as a function of the PCs with an associated estimation error. s η = f(η)+error () The function is created on a training data set where η is calculated from Equation 4 and s η is calculated from Equation 9 with ω k being calculated from the real values of X. The function f is then tested on a distinct testing sample from X. The current study analyzes five unique non-linear regression models, a simple linear regression model, the general additive model, multivariate adaptive regression splines, support vector regression, and the response manifold regression which is a new Gaussian-kernel based regression method. A brief mathematical description and explanation of these regression techniques follows. 4
5 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics - Linear Regression Model (LI) The linear model applied in multiple dimensions is of the form s η = ηa + v () Where a is the regression coefficient vector and v is the intercept vector []. - General Additive Model (GAM) The general additive model is a more rigorous concept of the linear model where instead of fitting a regression coefficient vector, functions are fitted in attempt to more accurately model the dependent variable. The general form of the model is where f η are functions dependent on η []. - Multivariate Adaptive Regression Splines (MARS) s η = f η η (2) Multivariate adaptive regression splines use the concept of building up the model from product spline basis functions. This model creates a number of basis functions, and automatically determines knot location and implements splines at knot boundaries. The model is of the form M s η = a m B m (η) (3) m= where B m are the basis functions and a m are the expansion coefficients [2]. - Support Vector Regression (SVR) Support vector regression is a subset of the support vector machine work. The idea behind SVR is again to create a model which predicts s η given η using learning machines which implement the structural risk minimization inductive principle. The basic model form is s η = (αi α i ) K (η,η i ) (4) i= where α i and α i are Lagrange multipliers, and K (η,η i ) is the kernel operator [3]. - Response Manifold (RM) The response manifold approach uses the concept of the adaraya-watson kernel estimation [4, 5]: K (η,η i ) s η,i i= s η = (5) K (η,η i ) i= where the kernel function K provides the highest weights to the neighboring data points giving a local estimation, being dependent on the selected filter width. η is the current value 5
6 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics of the test PCs, η i are the training PCs, s η,i are the training source-terms and K (η,η i ) is the kernel operator evaluated at the current η. The drawback to the adaraya-watson kernel estimation is the expensive evaluation of the kernel function when dealing with large data sets, or domains. The response manifold approach tabulates a manifold for the dependent variables the response based on a grid spanning the independent variable space. During run-time a quick interpolation provides the dependent variables from the tabulated response manifold. A kernel filter width, and manifold grid point spacing is selected so as to avoid over-fitting and provide accurate estimation of source-terms. 2.5 Data Set The training data set, used to create the non-linear regression models and for the PC analysis, must contain several important features. First, the data-set should contain realizations both varying temporally and spatially. Second the training data-set must produce the same low dimensional manifold in PC space as the combustion problem of interest. It has been demonstrated [8] that the low dimensional manifolds, may in fact be invariant under certain conditions, allowing a system to be modeled using the PCA from a similar combustion case. For demonstration of the regression process for use in the PC Transport mode, detailed in Section 2.3, a two-dimensional Direct umerical Simulation (DS) data set of a spherical premixed syn-gas air flame has been selected [6]. The detailed reaction scheme [7], contains 3 chemical species (CO, HCO, CH 2 O, CO 2, H 2 O, O 2, O, H, OH, HO 2, H 2 O 2, H 2, 2 ), and uses 67 chemical reactions. The DS is initialized with a unity equivalence ratio, with the fuel consisting of.5 CO and.5 H 2, air as an oxidizer, and a rms turbulent velocity fluctuation (u ) of m/s. The DS assumes a unity Lewis number, and has grid consisting of 8 by 8 points spanning.2 by.2 meters. The data set consists of 3 time-steps. 3 Results and Discussion Results using the various regression models outlined in Section 2.4 were computed with the statistical computing package R, and Matlab. The R code implementations for LI, GAM, MARS, and SVM were used. The authors RM model was implemented in Matlab. Sample training and testing data sets were taken randomly from the data set, where each sample consisted of, observations. The coefficient of determination (R 2 ) and the normalized root mean square error (RMSE) were used as a means of quantifying the error produced by the models: R 2 = RMSE = (x predicted,i x) 2 i= (6) (x i x) 2 i= (x predicted,i x i ) 2 i= max(σ(x predicted,x)) 6 (7)
7 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics.8.6 RAGE R η Figure : Mean R 2 from the reconstruction of the chemical species mass fractions, as a function of the number of retained η, using scaling. First the ability of PCA to represent the chemical species mass fractions is shown in Figures and 2. Figure shows R 2 is greater than.95 when η 4 (for scaling), and Figure 2 shows a RMSE value less than.85. As mentioned in Section 2.3 the propagation of error from the approximation of the state space is much more severe in the calculation of s η. Figures 3 and 4 show the error in the calculation of s η using Equation 9 directly. It is observed that when η 9 the coefficient of determination is great than.95 and a significant reduction in mean RMSE is observed. In order to avoid the calculation of the chemical species source-terms and to prohibit the large degree of error in the calculation of s η which is seen when using fewer η, the regression methods are now tested. Table shows the resultant error for the estimation of s η when four PCs are retained, regressing the value of s η on only three PCs. Table : R 2 for estimation of s η using scaling with various Regression Methods Regression Method s η, s η,2 s η,3 s η,4 LI GAM MARS SVM RM A dramatic improvement in the estimation of s η is observed. This dramatic improvement is consistent with the work by Biglari [8]. In particular the SVM, and RM regression methods show remarkable accuracy in estimating the source-terms. Another important factor in PCA is the scaling factor γ used in Equation 2. Figures 5 and 6 show the PCA manifold calculated using scaling and RAGE scaling respectively. The color in the Figures represent the value of the first PC source term s η,. Here one observes the difference in scales, shapes, and gradients the different 7
8 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics RMSE RAGE η Figure 2: Mean RMSE from the reconstruction of the chemical species mass fractions, as a function of the number of retained η, using scaling..8.6 RAGE R η Figure 3: Mean R 2 from the reconstruction of the source-terms for the species mass fractions, as a function of the number of retained η, using scaling. 8
9 8th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 2 RAGE RMSE η Figure 4: Mean RMSE from the reconstruction of sη, as a function of the number of retained η, using scaling. 6 x 2 η η2 4 η 2 Figure 5: PCA manifold created using scaling, here the independent variables are η, η2, and η3 representing the cartesian coordinates, and the dependent variable sη mapped in color to the manifold. scaling methods produce. Tables 2 and 3 show R2 and RMSE error metrics using the various scaling methods presented in Section 2.2 while using the RM regression method. Table 2: R2 for the estimation of sη using the RM regression method with various scaling methods Scaling Method RAGE sη, sη, sη, sη,
10 8th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics 4 x 2 η η2.5 η.5 Figure 6: PCA manifold created using RAGE scaling, here the independent variables are η, η2, and η3 representing the cartesian coordinates, and the dependent variable sη mapped in color to the manifold. Table 3: RMSE for the estimation of sη using the RM regression method with various scal- ing methods Scaling Method RAGE sη, sη, sη, sη, A clear benefit is seen while using the RM regression method with or scaling. This is due to that the fact that these manifolds contain smooth gradients, and a more simplified manifold shape allowing for a more accurate regression. 4 Conclusion The current work has addressed the ability to use non-linear regression methods to estimate sourceterms for a PCA based combustion model. Various non-linear regression methods have been analyzed showing the ability to produce accurate estimation even when using a lower number of η. In particular the SVM and RM methods have shown desired accuracy in estimation of sη. In addition the effect of various PCA scaling methods on the non-linear regression models have been assessed, with excellent results using and scaling. The current work outlines an example of an apriori analysis which provides the best regression and scaling method for a given turbulent combustion data set. Additional work may include an analysis on the invariance of the PC based manifold with respect to flow conditions, specifically by increasing the turbulence intensity, and a demonstration of the non-linear regression methods in conjunction with a simple perfectly stirred reactor system.
11 8 th US Combustion Meeting Paper # 7RK-252 Topic: Reaction Kinetics Acknowledgments We are grateful to our sponsor for which part of the present research was funded: The ational uclear Security Administration under the Accelerating Development of Retrofittable CO 2 Capture Technologies through Predictivity program through DOE Cooperative Agreement DE A 74. References [] Alessandro Parente, James C. Sutherland, L. Tognotti, and Philip J. Smith. Proceedings of the Combustion Institute, (29) journal. [2] J. C. Sutherland and A. Parente. Proceedings of the Combustion Institute, 32 (29) [3] G. P. Smith, D. M. Golden, M. Frenklach,. W. Moriarty, B. Eiteneer, M. Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, V. V. Lissianski, and Z. Qin journal. [4] R. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 23. [5] W.P. Jones and Rigopoulos Stelios. Combustion and Flame, 42 (25) [6]. Peters. Progress in Energy and Combustion Science, (984) [7] James C. Sutherland and Alessandro Parente. Proceedings of the Combustion Institute, 32 (29) [8] I. T. Jolliffe. Principal Component Analysis. Springer, ew York, Y, 986. [9] T. Poinsot and D. Veynante. Theoretical and umerical Combustion. R.T. Edwards, Inc., 2. [] William S Cleveland, Eric Grosse, and William M Shyu. Statistical models in S, (992) [] Simon Wood. Generalized additive models: an introduction with R, Vol. 66. Chapman & Hall/CRC, 26. [2] Jerome H Friedman. The annals of statistics, (99) 67. [3] Alex J Smola and Bernhard Schölkopf. Statistics and computing, 4 (24) [4] Elizbar A adaraya. Theory of Probability & Its Applications, 9 (964) [5] Geoffrey S Watson. Sankhyā: The Indian Journal of Statistics, Series A, (964) [6] Gordon Fru, Gábor Janiga, and Dominique Thévenin. Flow, turbulence and combustion, 88 (22) [7] Ulrich Maas and Stephen B Pope. Combustion and Flame, 88 (992) [8] Amir Biglari and James C Sutherland. Combustion and Flame, (22) journal.
Combustion Reaction Model Generation using Principal Component Analysis
Paper # 7F-6 Topic: Turbulent Combustion 7 Fall Meeting of the Western States Section of the Combustion Institute Sandia National Laboratories, Livermore, CA October 6 & 7, 7. Combustion Reaction Model
More informationFlamelet Analysis of Turbulent Combustion
Flamelet Analysis of Turbulent Combustion R.J.M. Bastiaans,2, S.M. Martin, H. Pitsch,J.A.vanOijen 2, and L.P.H. de Goey 2 Center for Turbulence Research, Stanford University, CA 9435, USA 2 Eindhoven University
More informationThermal NO Predictions in Glass Furnaces: A Subgrid Scale Validation Study
Feb 12 th 2004 Thermal NO Predictions in Glass Furnaces: A Subgrid Scale Validation Study Padmabhushana R. Desam & Prof. Philip J. Smith CRSIM, University of Utah Salt lake city, UT-84112 18 th Annual
More informationREDIM reduced modeling of quenching at a cold inert wall with detailed transport and different mechanisms
26 th ICDERS July 3 th August 4 th, 217 Boston, MA, USA REDIM reduced modeling of quenching at a cold inert wall with detailed transport and different mechanisms Christina Strassacker, Viatcheslav Bykov,
More informationIdentification of low-dimensional manifolds in turbulent flames
Available online at www.sciencedirect.com Proceedings of the Combustion Institute 32 (2009) 1579 1586 Proceedings of the Combustion Institute www.elsevier.com/locate/proci Identification of low-dimensional
More informationCombustion modeling using principal component analysis
Available online at www.sciencedirect.com Proceedings of the Combustion Institute 32 (2009) 1563 1570 Proceedings of the Combustion Institute www.elsevier.com/locate/proci Combustion modeling using principal
More informationA comparison between two different Flamelet reduced order manifolds for non-premixed turbulent flames
8 th U. S. National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013 A comparison between two different Flamelet
More informationPredicting NO Formation with Flamelet Generated Manifolds
Predicting NO Formation with Flamelet Generated Manifolds J. A. van Oijen and L. P. H. de Goey Dept. Mechanical Engineering, Technische Universiteit Eindhoven P.O. Box, 6 MB Eindhoven, The Netherlands
More informationNumerical Investigation of Ignition Delay in Methane-Air Mixtures using Conditional Moment Closure
21 st ICDERS July 23-27, 27 Poitiers, France Numerical Investigation of Ignition Delay in Methane-Air Mixtures using Conditional Moment Closure Ahmad S. El Sayed, Cécile B. Devaud Department of Mechanical
More informationCH 4 /NO x Reduced Mechanisms Used for Modeling Premixed Combustion
Energy and Power Engineering, 2012, 4, 264-273 http://dx.doi.org/10.4236/epe.2012.44036 Published Online July 2012 (http://www.scirp.org/journal/epe) CH 4 /NO x Reduced Mechanisms Used for Modeling Premixed
More informationOn Computing Reactive Flows Using Slow Manifolds
On Computing Reactive Flows Using Slow Manifolds by S. Paolucci 1, J. M. Powers 2, and S. Singh 3 Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana 46556-5637,
More informationTowards regime identification and appropriate chemistry tabulation for computation of autoigniting turbulent reacting flows
Center for Turbulence Research Annual Research Briefs 009 199 Towards regime identification and appropriate chemistry tabulation for computation of autoigniting turbulent reacting flows By M. Kostka, E.
More informationA REDUCED-ORDER METHANE-AIR COMBUSTION MECHANISM THAT SATISFIES THE DIFFERENTIAL ENTROPY INEQUALITY
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Special Issue/2018, pp. 285 290 A REDUCED-ORDER METHANE-AIR COMBUSTION MECHANISM THAT SATISFIES THE DIFFERENTIAL
More informationMulti-dimensional transport: DNS analysis and incorporation into the Reaction-Diffusion Manifold (REDIM) method
25 th ICDERS August 2 7, 2015 Leeds, UK Multi-dimensional transport: DNS analysis and incorporation into the Reaction-Diffusion Manifold (REDIM) method R. Schießl, V. Bykov and U. Maas Institut für Technische
More informationUQ in Reacting Flows
UQ in Reacting Flows Planetary Entry Simulations High-Temperature Reactive Flow During descent in the atmosphere vehicles experience extreme heating loads The design of the thermal protection system (TPS)
More informationHierarchical approach
Chemical mechanisms Examine (i) ways in which mechanisms are constructed, (ii)their dependence on rate and thermodynamic data and (iii) their evaluation using experimental targets Copyright 2011 by Michael
More informationConnection of Local Linear Embedding, ISOMAP, and Kernel Principal Component Analysis
Connection of Local Linear Embedding, ISOMAP, and Kernel Principal Component Analysis Alvina Goh Vision Reading Group 13 October 2005 Connection of Local Linear Embedding, ISOMAP, and Kernel Principal
More informationA Priori Model for the Effective Lewis Numbers in Premixed Turbulent Flames
Paper # 070LT-0267 Topic: Turbulent Flames 8 th US National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013.
More informationLaminar Premixed Flames: Flame Structure
Laminar Premixed Flames: Flame Structure Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Course Overview Part I: Fundamentals and Laminar Flames Introduction Fundamentals and mass balances of
More informationVALIDATION OF THE ASVDADD CONSTRAINT SELECTION ALGORITHM FOR EFFECTIVE RCCE MODELING OF NATURAL GAS IGNITION IN AIR
Proceedings of the ASME 2016 International Mechanical Engineering Congress & Exposition IMECE2016 November 11-17, 2016, Phoenix, Arizona, USA IMECE2016-65323 VALIDATION OF THE ASVDADD CONSTRAINT SELECTION
More informationIntroduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones
Introduction to machine learning and pattern recognition Lecture 2 Coryn Bailer-Jones http://www.mpia.de/homes/calj/mlpr_mpia2008.html 1 1 Last week... supervised and unsupervised methods need adaptive
More informationThe role of diffusion at shear layers in irregular detonations
The role of diffusion at shear layers in irregular detonations Marco Arienti 1 Joseph E. Shepherd 2 1 United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108 2 California Institute
More informationKINETIC MODELING OF OXIDATION METHANE CONVERSION IN REGIME OF FILTRATION COMBUSTION WITH SUPERADIABATIC HEATING
KINETIC MODELING OF OXIDATION METHANE CONVERSION IN REGIME OF FILTRATION COMBUSTION WITH SUPERADIABATIC HEATING Anna A. Karnaukh, Avigeya N. Ivanova, Svetlana S. Kostenko, George B. Manelis, and Eugene
More informationGradient Enhanced Universal Kriging Model for Uncertainty Propagation in Nuclear Engineering
Gradient Enhanced Universal Kriging Model for Uncertainty Propagation in Nuclear Engineering Brian A. Lockwood 1 and Mihai Anitescu 2 1 Department of Mechanical Engineering University of Wyoming 2 Mathematics
More informationDiscriminative Direction for Kernel Classifiers
Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering
More informationIMPROVED POLLUTANT PREDICTIONS IN LARGE-EDDY SIMULATIONS OF TURBULENT NON-PREMIXED COMBUSTION BY CONSIDERING SCALAR DISSIPATION RATE FLUCTUATIONS
Proceedings of the Combustion Institute, Volume 9, 00/pp. 1971 1978 IMPROVED POLLUTANT PREDICTIONS IN LARGE-EDDY SIMULATIONS OF TURBULENT NON-PREMIXED COMBUSTION BY CONSIDERING SCALAR DISSIPATION RATE
More informationAsymptotic Structure of Rich Methane-Air Flames
Asymptotic Structure of Rich Methane-Air Flames K. SESHADRI* Center for Energy and Combustion Research, Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla,
More informationTAU Extensions for High Enthalpy Flows. Sebastian Karl AS-RF
TAU Extensions for High Enthalpy Flows Sebastian Karl AS-RF Contents Motivation Extensions available in the current release: Numerical schemes for super- and hypersonic flow fields Models for gas mixtures,
More informationPremixed MILD Combustion of Propane in a Cylindrical. Furnace with a Single Jet Burner: Combustion and. Emission Characteristics
Premixed MILD Combustion of Propane in a Cylindrical Furnace with a Single Jet Burner: Combustion and Emission Characteristics Kin-Pang Cheong a, c, Guochang Wang a, Jianchun Mi a*, Bo Wang a, Rong Zhu
More informationLecture 6 Asymptotic Structure for Four-Step Premixed Stoichiometric Methane Flames
Lecture 6 Asymptotic Structure for Four-Step Premixed Stoichiometric Methane Flames 6.-1 Previous lecture: Asymptotic description of premixed flames based on an assumed one-step reaction. basic understanding
More informationUnsupervised Learning: Dimensionality Reduction
Unsupervised Learning: Dimensionality Reduction CMPSCI 689 Fall 2015 Sridhar Mahadevan Lecture 3 Outline In this lecture, we set about to solve the problem posed in the previous lecture Given a dataset,
More informationMachine learning for pervasive systems Classification in high-dimensional spaces
Machine learning for pervasive systems Classification in high-dimensional spaces Department of Communications and Networking Aalto University, School of Electrical Engineering stephan.sigg@aalto.fi Version
More informationComputation. For QDA we need to calculate: Lets first consider the case that
Computation For QDA we need to calculate: δ (x) = 1 2 log( Σ ) 1 2 (x µ ) Σ 1 (x µ ) + log(π ) Lets first consider the case that Σ = I,. This is the case where each distribution is spherical, around the
More informationApproximation of chemical reaction rates in turbulent combustion simulation
Approximation of chemical reaction rates in turbulent combustion simulation Lars Frank Große and Franz Joos * Helmut-Schmidt-University University of the Federal Armed Forces Hamburg - Laboratory of Turbo
More informationPREDICTING SOLAR GENERATION FROM WEATHER FORECASTS. Chenlin Wu Yuhan Lou
PREDICTING SOLAR GENERATION FROM WEATHER FORECASTS Chenlin Wu Yuhan Lou Background Smart grid: increasing the contribution of renewable in grid energy Solar generation: intermittent and nondispatchable
More informationUTSR Fellowship Presentation Gas Turbine Industrial Fellowship Program 2006
UTSR Fellowship Presentation Gas Turbine Industrial Fellowship Program 2006 Predicting Lean Blowout Using the Damkohler Number Matthew J. Bloxham, Brigham Young University Ingersoll Rand Energy Systems
More informationBest Practice Guidelines for Combustion Modeling. Raphael David A. Bacchi, ESSS
Best Practice Guidelines for Combustion Modeling Raphael David A. Bacchi, ESSS PRESENTATION TOPICS Introduction; Combustion Phenomenology; Combustion Modeling; Reaction Mechanism; Radiation; Case Studies;
More informationFlow and added small-scale topologies in a turbulent premixed flame
Flow and added small-scale topologies in a turbulent premixed flame L. Cifuentes*, A. Kempf* and C. Dopazo** luis.cifuentes@uni-due.de *University of Duisburg-Essen, Chair of Fluid Dynamics, Duisburg -
More informationRegularization via Spectral Filtering
Regularization via Spectral Filtering Lorenzo Rosasco MIT, 9.520 Class 7 About this class Goal To discuss how a class of regularization methods originally designed for solving ill-posed inverse problems,
More informationEfficient and accurate time-integration of combustion chemical kinetics using artificial neural networks
Efficient and accurate time-integration of combustion chemical kinetics using artificial neural networks Wen Yu Peng (wypeng), Nicolas H. Pinkowski (npinkows) Abstract An artificial neural network (ANN)
More informationANSYS Advanced Solutions for Gas Turbine Combustion. Gilles Eggenspieler 2011 ANSYS, Inc.
ANSYS Advanced Solutions for Gas Turbine Combustion Gilles Eggenspieler ANSYS, Inc. 1 Agenda Steady State: New and Existing Capabilities Reduced Order Combustion Models Finite-Rate Chemistry Models Chemistry
More informationDARS overview, IISc Bangalore 18/03/2014
www.cd-adapco.com CH2O Temperatur e Air C2H4 Air DARS overview, IISc Bangalore 18/03/2014 Outline Introduction Modeling reactions in CFD CFD to DARS Introduction to DARS DARS capabilities and applications
More informationEFFECTS OF PRESSURE AND PREHEAT ON SUPER-ADIABATIC FLAME TEMPERATURES IN RICH PREMIXED METHANE/AIR FLAMES
Combust. Sci. and Tech., 180: 437 452, 2008 Copyright # Taylor & Francis Group, LLC ISSN: 0010-2202 print/1563-521x online DOI: 10.1080/00102200701741285 EFFECTS OF PRESSURE AND PREHEAT ON SUPER-ADIABATIC
More informationD. VEYNANTE. Introduction à la Combustion Turbulente. Dimanche 30 Mai 2010, 09h00 10h30
D. VEYNANTE Introduction à la Combustion Turbulente Dimanche 30 Mai 2010, 09h00 10h30 Introduction to turbulent combustion D. Veynante Laboratoire E.M2.C. CNRS - Ecole Centrale Paris Châtenay-Malabry France
More informationarxiv: v1 [physics.chem-ph] 6 Oct 2011
Calculation of the Minimum Ignition Energy based on the ignition delay time arxiv:1110.1163v1 [physics.chem-ph] 6 Oct 2011 Jens Tarjei Jensen a, Nils Erland L. Haugen b, Natalia Babkovskaia c a Department
More informationA G-equation formulation for large-eddy simulation of premixed turbulent combustion
Center for Turbulence Research Annual Research Briefs 2002 3 A G-equation formulation for large-eddy simulation of premixed turbulent combustion By H. Pitsch 1. Motivation and objectives Premixed turbulent
More informationSpectral Regularization
Spectral Regularization Lorenzo Rosasco 9.520 Class 07 February 27, 2008 About this class Goal To discuss how a class of regularization methods originally designed for solving ill-posed inverse problems,
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Principal Components Analysis (PCA) a technique for finding patterns in data of high dimension Outline:. Eigenvectors and eigenvalues. PCA: a) Getting the data b) Centering
More informationDNS and LES of Turbulent Combustion
Computational Fluid Dynamics In Chemical Reaction Engineering IV June 19-24, 2005 Barga, Italy DNS and LES of Turbulent Combustion Luc Vervisch INSA de Rouen, IUF, CORIA-CNRS Pascale Domingo, Julien Réveillon
More informationSynergy between Data Reconciliation and Principal Component Analysis.
Plant Monitoring and Fault Detection Synergy between Data Reconciliation and Principal Component Analysis. Th. Amand a, G. Heyen a, B. Kalitventzeff b Thierry.Amand@ulg.ac.be, G.Heyen@ulg.ac.be, B.Kalitventzeff@ulg.ac.be
More informationExercises in Combustion Technology
Exercises in Combustion Technology Exercise 4: Turbulent Premixed Flames Turbulent Flow: Task 1: Estimation of Turbulence Quantities Borghi-Peters diagram for premixed combustion Task 2: Derivation of
More informationA priori Tabulation of Turbulent Flame Speeds via a Combination of a Stochastic Mixing Model and Flamelet Generated Manifolds 5
Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany HEIKO SCHMIDT 1 MICHAEL OEVERMANN 2 ROB J.M. BASTIAANS 3 ALAN R. KERSTEIN 4 A priori Tabulation of Turbulent
More informationSimulation of Turbulent Lifted Flames and their Transient Propagation
25 th ICDERS August 2-7th, 2015 Leeds, UK Simulation of Turbulent Lifted Flames and their Transient Propagation S. Ruan, Z. Chen, N. Swaminathan University of Cambridge Cambridge, UK 1 Introduction Turbulent
More informationSkeletal Kinetic Mechanism of Methane Oxidation for High Pressures and Temperatures
7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS) Skeletal Kinetic Mechanism of Methane Oxidation for High Pressures and Temperatures Victor P. Zhukov and Alan F. Kong Institute of Space
More informationA Priori Testing of Flamelet Generated Manifolds for Turbulent Partially Premixed Methane/Air Flames
DOI 10.1007/s10494-009-9223-1 A Priori Testing of Flamelet Generated Manifolds for Turbulent Partially Premixed Methane/Air Flames W. J. S. Ramaekers J. A. van Oijen L. P. H. de Goey Received: 7 September
More informationMauro Valorani Dipartimento di Meccanica e Aeronautica, University of Rome
Classification of ignition regimes in thermally stratified n-heptane-air air mixtures using computational singular perturbation Saurabh Gupta, Hong G. Im Department of Mechanical Engineering, University
More informationExtinction Limits of Premixed Combustion Assisted by Catalytic Reaction in a Stagnation-Point Flow
44th AIAA Aerospace Sciences Meeting and Exhibit 9-12 January 2006, Reno, Nevada AIAA 2006-164 Extinction Limits of Premixed Combustion Assisted by Catalytic Reaction in a Stagnation-Point Flow Jingjing
More informationSuper-adiabatic flame temperatures in premixed methane-oxygen flames
Super-adiabatic flame temperatures in premixed methane-oxygen flames Björn Stelzner, Christof Weis, Peter Habisreuther, Nikolaos Zarzalis, Dimosthenis Trimis Karlsruhe Institute of Technology, Engler-Bunte-Institute,
More informationLeast Absolute Shrinkage is Equivalent to Quadratic Penalization
Least Absolute Shrinkage is Equivalent to Quadratic Penalization Yves Grandvalet Heudiasyc, UMR CNRS 6599, Université de Technologie de Compiègne, BP 20.529, 60205 Compiègne Cedex, France Yves.Grandvalet@hds.utc.fr
More information1 Principal Components Analysis
Lecture 3 and 4 Sept. 18 and Sept.20-2006 Data Visualization STAT 442 / 890, CM 462 Lecture: Ali Ghodsi 1 Principal Components Analysis Principal components analysis (PCA) is a very popular technique for
More informationModeling instabilities in lean premixed turbulent combustors using detailed chemical kinetics
Accepted for publication in Combustion Science and Technology Modeling instabilities in lean premixed turbulent combustors using detailed chemical kinetics Bjørn Lilleberg, Ivar S. Ertesvåg and Kjell Erik
More informationCFD and Kinetic Analysis of Bluff Body Stabilized Flame
CFD and Kinetic Analysis of Bluff Body Stabilized ame A. Dicorato, E. Covelli, A. Frassoldati, T. Faravelli, E. Ranzi Dipartimento di Chimica, Materiali e Ingegneria Chimica, Politecnico di Milano, ITALY
More informationIntroduction to Machine Learning
10-701 Introduction to Machine Learning PCA Slides based on 18-661 Fall 2018 PCA Raw data can be Complex, High-dimensional To understand a phenomenon we measure various related quantities If we knew what
More informationPrincipal Component Analysis-I Geog 210C Introduction to Spatial Data Analysis. Chris Funk. Lecture 17
Principal Component Analysis-I Geog 210C Introduction to Spatial Data Analysis Chris Funk Lecture 17 Outline Filters and Rotations Generating co-varying random fields Translating co-varying fields into
More informationExamination of the effect of differential molecular diffusion in DNS of turbulent non-premixed flames
Examination of the effect of differential molecular diffusion in DNS of turbulent non-premixed flames Chao Han a, David O. Lignell b, Evatt R. Hawkes c, Jacqueline H. Chen d, Haifeng Wang a, a School of
More informationInvestigation of ignition dynamics in a H2/air mixing layer with an embedded vortex
Paper # 070LT-0211 The 8th US National Meeting of the Combustion Institute, Park City, UT, May 19-22, 2013 Investigation of ignition dynamics in a H2/air mixing layer with an embedded vortex S.K. Menon
More informationPrincipal Component Analysis (PCA) of AIRS Data
Principal Component Analysis (PCA) of AIRS Data Mitchell D. Goldberg 1, Lihang Zhou 2, Walter Wolf 2 and Chris Barnet 1 NOAA/NESDIS/Office of Research and Applications, Camp Springs, MD 1 QSS Group Inc.
More informationExperimental study of the combustion properties of methane/hydrogen mixtures Gersen, Sander
University of Groningen Experimental study of the combustion properties of methane/hydrogen mixtures Gersen, Sander IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF)
More informationHale Collage. Spectropolarimetric Diagnostic Techniques!!!!!!!! Rebecca Centeno
Hale Collage Spectropolarimetric Diagnostic Techniques Rebecca Centeno March 1-8, 2016 Today Simple solutions to the RTE Milne-Eddington approximation LTE solution The general inversion problem Spectral
More informationOn two fractional step finite volume and finite element schemes for reactive low Mach number flows
Fourth International Symposium on Finite Volumes for Complex Applications - Problems and Perspectives - July 4-8, 2005 / Marrakech, Morocco On two fractional step finite volume and finite element schemes
More informationKernel Methods. Machine Learning A W VO
Kernel Methods Machine Learning A 708.063 07W VO Outline 1. Dual representation 2. The kernel concept 3. Properties of kernels 4. Examples of kernel machines Kernel PCA Support vector regression (Relevance
More informationDirect numerical prediction of OH-LIF Signals in the Simulation of a laminar partial oxidation flame
Direct numerical prediction of OH-LIF Signals in the Simulation of a laminar partial oxidation flame F. Hunger 1, B. Stelzner 2, D. Trimis 2, C. Hasse 1 1 Chair of Numerical Thermo-Fluid Dynamics, ZIK
More informationMACHINE LEARNING. Methods for feature extraction and reduction of dimensionality: Probabilistic PCA and kernel PCA
1 MACHINE LEARNING Methods for feature extraction and reduction of dimensionality: Probabilistic PCA and kernel PCA 2 Practicals Next Week Next Week, Practical Session on Computer Takes Place in Room GR
More informationDynamics of Excited Hydroxyl Radicals in Hydrogen Based Mixtures Behind Reflected Shock Waves. Supplemental material
Dynamics of Excited Hydroxyl Radicals in Hydrogen Based Mixtures Behind Reflected Shock Waves Proceedings of the Combustion Institute 34, 22 Supplemental material R. MÉVEL, S. PICHON, L. CATOIRE, N. CHAUMEIX,
More informationCombustion. Indian Institute of Science Bangalore
Combustion Indian Institute of Science Bangalore Combustion Applies to a large variety of natural and artificial processes Source of energy for most of the applications today Involves exothermic chemical
More informationEINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science. CASA-Report March2008
EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science CASA-Report 08-08 March2008 The complexe flux scheme for spherically symmetrie conservation laws by J.H.M. ten Thije Boonkkamp,
More informationTutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and
More informationMachine Learning. B. Unsupervised Learning B.2 Dimensionality Reduction. Lars Schmidt-Thieme, Nicolas Schilling
Machine Learning B. Unsupervised Learning B.2 Dimensionality Reduction Lars Schmidt-Thieme, Nicolas Schilling Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Feature Extraction Hamid R. Rabiee Jafar Muhammadi, Alireza Ghasemi, Payam Siyari Spring 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2/ Agenda Dimensionality Reduction
More informationDevelopment of Constrained Equilibrium Codes and Their Applications in Nonequilibrium Thermodynamics
Partha S. Bishnu Djamel Hamiroune Mohamad Metghalchi e-mail: metghal@coe.neu.edu Mechanical, Industrial and Manufacturing Engineering Department, Northeastern University, Boston, MA 02115 Development of
More informationModeling of Wall Heat Transfer and Flame/Wall Interaction A Flamelet Model with Heat-Loss Effects
9 th U. S. National Combustion Meeting Organized by the Central States Section of the Combustion Institute May 17-20, 2015 Cincinnati, Ohio Modeling of Wall Heat Transfer and Flame/Wall Interaction A Flamelet
More informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
More informationLinear Methods for Regression. Lijun Zhang
Linear Methods for Regression Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline Introduction Linear Regression Models and Least Squares Subset Selection Shrinkage Methods Methods Using Derived
More informationUNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2013
UNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2013 Exam policy: This exam allows two one-page, two-sided cheat sheets; No other materials. Time: 2 hours. Be sure to write your name and
More informationDevelopment of Reduced Mechanisms for Numerical Modelling of Turbulent Combustion
Worshop on Numerical Aspects of Reduction in Chemical Kinetics CERMICS-ENPC Cite Descartes - Champus sur Marne, France, September 2nd, 1997 Abstract Development of Reduced Mechanisms for Numerical Modelling
More informationFlame / wall interaction and maximum wall heat fluxes in diffusion burners
Flame / wall interaction and maximum wall heat fluxes in diffusion burners de Lataillade A. 1, Dabireau F. 1, Cuenot B. 1 and Poinsot T. 1 2 June 5, 2002 1 CERFACS 42 Avenue Coriolis 31057 TOULOUSE CEDEX
More information2D Direct Numerical Simulation of methane/air turbulent premixed flames under high turbulence intensity Julien Savre 04/13/2011
1 2D Direct Numerical Simulation of methane/air turbulent premixed flames under high turbulence intensity Julien Savre 04/13/2011 2 Outline Why studying turbulent premixed flames under high turbulent intensity?
More informationCombustion basics... We are discussing gaseous combustion in a mixture of perfect gases containing N species indexed with k=1 to N:
Combustion basics... T. Poinsot poinsot@imft.fr Only the things you should know to understand the following courses Mainly elements of laminar flame theory for premixed and diffusion flames 1 Copyright
More informationChemistry 123: Physical and Organic Chemistry Topic 4: Gaseous Equilibrium
Topic 4: Introduction, Topic 4: Gaseous Equilibrium Text: Chapter 6 & 15 4.0 Brief review of Kinetic theory of gasses (Chapter 6) 4.1 Concept of dynamic equilibrium 4.2 General form & properties of equilbrium
More informationAn investigation of the performance of turbulent mixing models
Combustion and Flame 136 (24) 28 216 www.elsevier.com/locate/jnlabr/cnf An investigation of the performance of turbulent mixing models Zhuyin Ren and Stephen B. Pope Sibley School of Mechanical and Aerospace
More informationThe Effect of Flame Structure on Soot Formation and Transport in Turbulent Nonpremixed Flames Using Direct Numerical Simulation
The Effect of Flame Structure on Soot Formation and Transport in Turbulent Nonpremixed Flames Using Direct Numerical Simulation David O. Lignell a,, Jacqueline H. Chen a, Philip J. Smith b, Tianfeng F.
More informationCombustion and Flame. Direct numerical simulation of auto-ignition of a hydrogen vortex ring reacting with hot air
Combustion and Flame 156 (2009) 813 825 Contents lists available at ScienceDirect Combustion and Flame www.elsevier.com/locate/combustflame Direct numerical simulation of auto-ignition of a hydrogen vortex
More informationFluid Dynamics and Balance Equations for Reacting Flows
Fluid Dynamics and Balance Equations for Reacting Flows Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Balance Equations Basics: equations of continuum mechanics balance equations for mass and
More informationGaussian Process Approximations of Stochastic Differential Equations
Gaussian Process Approximations of Stochastic Differential Equations Cédric Archambeau Centre for Computational Statistics and Machine Learning University College London c.archambeau@cs.ucl.ac.uk CSML
More informationInvestigation of ignition dynamics in a H2/air mixing layer with an embedded vortex
Paper # 070LT-0211 The 8th US National Meeting of the Combustion Institute, Park City, UT, May 19-22, 2013 Investigation of ignition dynamics in a H2/air mixing layer with an embedded vortex S.K. Menon
More informationLearning sets and subspaces: a spectral approach
Learning sets and subspaces: a spectral approach Alessandro Rudi DIBRIS, Università di Genova Optimization and dynamical processes in Statistical learning and inverse problems Sept 8-12, 2014 A world of
More informationNumerical Simulation of Entropy Generation in Hydrogen Enriched Swirl Stabilized Combustion
Saqr & Wahid CFD Letters Vol. 5(1) 13 www.cfdl.issres.net Vol. 5 (1) March 13 Numerical Simulation of Entropy Generation in Hydrogen Enriched Swirl Stabilized Combustion Khalid M. Saqr 1,* and Mazlan A.
More informationCHAPTER 4 PRINCIPAL COMPONENT ANALYSIS-BASED FUSION
59 CHAPTER 4 PRINCIPAL COMPONENT ANALYSIS-BASED FUSION 4. INTRODUCTION Weighted average-based fusion algorithms are one of the widely used fusion methods for multi-sensor data integration. These methods
More informationCombustion and Emission Modeling in CONVERGE with LOGE models
Combustion and Emission Modeling in CONVERGE with LOGE models Corinna Netzer, Harry Lehtiniemi and Fabian Mauss 2015 CONVERGE USER CONFERENCE RICHARD CHILDRESS RACING, WELCOME, NC Outline Objective LOGE
More informationLecture 8 Laminar Diffusion Flames: Diffusion Flamelet Theory
Lecture 8 Laminar Diffusion Flames: Diffusion Flamelet Theory 8.-1 Systems, where fuel and oxidizer enter separately into the combustion chamber. Mixing takes place by convection and diffusion. Only where
More information